SEVENTH FRAMEWORK PROGRAMME
THEME [SST.2010.1.3-1.]
[Transport modelling for policy impact assessments]
Grant agreement for: Coordination and support action
Acronym: Transtools 3
Full title: „Research and development of the European Transport Network Model – Transtools
Version 3
Proposal/Contract no.: MOVE/FP7/266182/TRANSTOOLS 3
Start date: 1st March 2011
Duration: 36 months
MS.51 - “Passenger model design report”
Document number: TT3_WP8_MS51_TECH_Passenger model design report_0a
Workpackage: WP8
Deliverable nature: N/A
Dissemination level: N/A
Lead beneficiary: KTH (3), Svante Berglund
Due data of deliverable: Sept. 2011
Date of preparation of deliverable: 11. Sept. 2011
Date of last change: 22. Sept. 2011
Date of approval by Commission: N/A
Abstract:
This report is a description of the new passenger models that will be developed in Transtools 3.
Two models will be developed one short distance model for trips shorter than 100 km and one
model for trips longer than 100 km. The models will cover all relevant modes and trip purposes. In
the report we discuss availability of data, the limitations and possibilities in that respect. We also
outline the formulations for both models where we pay special attention to non linear utility
functions which is of great importance for models at a geographical scale of Europe. A key input
in travel demand comes from models of car ownership. Currently there is no model available at
the European level which surprised us. Beside the crucial importance of car ownership in models
of travel demand car ownership is subject to policy decisions at different levels. The rate of
growth in car ownership in Europe, particularly in regions where the economy grows from low
levels, got a potential to be a very important change in the European transport system.
Keywords:
Short distance trips, Long distance trips, Travel demand models, ETIS+, Travel data, car
ownership, Non linear utility functions
Author(s):
[Algers, Staffan], [Berglund, Svante]
Disclaimer:
The contents of this report reflect the views of the author and do not necessarily reflect the official
views or policy of the European Union. The European Union is not liable for any use that may be
made of the information contained in the report.
The report is not an official deliverable under the TT3 project and has not been reviewed or
approved by the Commission. The report is a working document of the Consortium.
MS51: Passenger model design report
Report version 0b
2011
By
Svante Berglund, KTH.
Copyright:
Published by:
Reproduction of this publication in whole or in part must include the customary
bibliographic citation, including author attribution, report title, etc.
Department of Transport, Bygningstorvet 116 Vest, DK-2800 Kgs. Lyngby,
Denmark
Request report
from:
www.transport.dtu.dk
Content
Summary ........................................................................................................................................6
1.1 Data requirements ...............................................................................................................6
1.2 Short distance model ...........................................................................................................6
1.3 Long distance model ............................................................................................................7
2.
2.1
2.2
Introduction ..........................................................................................................................8
Objective of the deliverable ..................................................................................................8
Methodology.........................................................................................................................8
3.
3.1
3.2
3.3
3.4
3.5
Data requirements ...............................................................................................................9
Data to support estimation - long distance trips ...................................................................9
Data to support base case application .................................................................................9
Data to support estimation - short distance trips ...............................................................10
Zoning system adjustment .................................................................................................10
Process ETIS+ (Rewrite wrt to what we will get from Otto concerning variables etc) .......10
4.
4.1
4.2
4.3
4.4
Modelling approach – some general issues ......................................................................11
Car ownership ....................................................................................................................11
Cost sharing car driver/car passenger ...............................................................................15
Vehicle fleet composition ...................................................................................................15
Work trip cost deductions ...................................................................................................15
5.
5.1
5.2
Short distance model .........................................................................................................16
Scope .................................................................................................................................16
Calibration ..........................................................................................................................18
6.
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Long distance models ........................................................................................................20
Data ....................................................................................................................................20
Segmentation by trip purpose ............................................................................................20
Seasonal variation of demand ...........................................................................................21
Geographical segmentation ...............................................................................................22
Non infrastructure network variables - Barriers and affinities ............................................23
Trip duration .......................................................................................................................23
Trip generation modelling .....................................................................................................24
Model for access and egress trips. ....................................................................................25
7.
7.1
7.2
Estimation procedures .......................................................................................................26
Short distance trips ............................................................................................................26
Long distance trips .............................................................................................................26
Appendix 1: Using Box-Cox approximations to estimate nonlinear utility functions in
discrete choice models ......................................................................................................29
Background ..................................................................................................................................29
MS51: Passenger model design report
Definition ......................................................................................................................................30
Possible approximations ..............................................................................................................30
Using approximations in real applications ....................................................................................37
References ...................................................................................................................................40
Appendix 2 Equivalence of Box-Cox and ‘Gamma’ functions ......................................................41
MS51: Passenger model design report
Summary
The scope of WP8 is to further develop and refine the passenger models from Transtools2. The
modelling work will consist of development of two models: one for short distance trips below 100 km
and one for long distance trips above 100 km. The scope of these two models is summarised below
together with data requirements. A previous version of this document has been discussed during
meetings at DTU and comments and suggestions has as far as possible been added to the current
document.
1.1 Data requirements
Model estimation will be separate for short distance trips and for long distance trips. For Long distance
trips, we will mainly rely on the Dateline data source from 2000 for observed travel behaviour. In order
to maintain consistency between dependent data (observed behaviour) and independent data (Level
of Service (LOS) and land use (LU)) that represents the situation in year 2000 is needed for the
estimation of long distance models. Background data will be delivered by an ongoing project ETIS+.
For long distance trips, the following data are needed from ETIS+:


LOS variables. LOS data consists of two components: network data (the infrastructure) and
the traffic provided on the infrastructure i.e. travel times, frequency and capacity (?).
LU variables are variables describing the attraction of each zone as destination for trips.
According to the ETIS+ and Transtools3 DOW’s, base matrices for short distance trips by modes and
travel purposes will be established. These matrices can be used as aggregate demand. For this,
population data by category, LOS and LU data are needed for the base case year. This includes data
for zone internal trips. This data will also support the base case application.
1.2 Short distance model
The scope is to establish a set of models by travel purpose containing mode, destination and
frequency choice. The model should be able to respond to transport policy changes with respect to
infrastructure and pricing as well as to background variables like land use, economic growth and
changes in car ownership. Car ownership is a key variable in travel demand models and care is
needed in formulation these variables. Somewhat surprisingly there is no forecasting model for car
ownership of reasonable quality available for Europe in the TREMOVE/SCENES framework.
The short distance model will use the following modes: car as driver, car as passenger, public
transport, slow modes (walk and bike) will be treated in a simplistic way. The available zone size and
the European scale are the reasons for the treatment of slow modes. The short distance model will
consist of the following trip types: Commuting, business, private and holiday trips.
6
1.3 Long distance model
The long distance models will have a similar structure as the short distance model but there will be
some differences. The segmentation in the long distance model will differ from the short distance
model and most likely will the utility functions differ from the short distance model where non linear
utility functions will be important in the long distance model. For the long distance model the
DATELINE survey from 2000 will be an important source of observed travel behaviour. The long
distance model will use five modes: air, car as driver, car as passenger, bus and rail. Available trip
purposes in the long distance model will be: holiday, business and private trips. Commuting trips could
either be present in both in the long and short distance model or we could extend the range of
commuting trips in the short distance model above 100 km. The final choice in this issue will be based
on empirical and technical considerations.
7
2. Introduction
The scope of WP8 is to establish a set of passenger demand models by travel purpose containing
mode, destination and frequency choice. The model should be able to respond to transport policy
changes with respect to infrastructure and pricing as well as to changes in background variables like
land use, investment schemes, economic growth and car ownership developments. The models that
will be the outcome of WP8 will be one central part of Transtools 3.
2.1 Objective of the deliverable
The model design report is the manuscript to the future work in WP8. We discuss important aspects of
the passenger demand models that will be developed within the WP and identifies possible problems
that may be of importance for the future work. Since the different WPs in Transtools is highly
depending on each other the report serves as an information to other WPs of what will be delivered
and what input the system must provide the passenger models with.
In the model design report we also raise some questions regarding supporting models (e.g. car
ownership) absence of important roles (tax deduction) and issues concerning model formulation.
Some of these questions are of empirical nature and will be solved during the project and some are a
matter of resources.
2.2 Methodology
The model design report is the outcome of an iterative process between the staff of WP8 and other
participants in the Transtools project. This report has been discussed during two expert meetings at
DTU (spring 2011 and autumn 2011) and suggestions and comments have been included in the final
version. During the process we have discussed different data sources and their merits and
shortcomings. The passenger demand models will end up as a part within the Transtools framework
and ongoing process is to create an understanding of how the pieces should fit together.
8
3. Data requirements
The data requirements need to serve two purposes: one is to support model estimation, and the other
one is to support generation of a base year application.
3.1 Data to support estimation - long distance trips
Model estimation will be separate for short distance trips and for long distance trips, possibly with the
exception of commute trips. For Long distance trips, we will mainly rely on the Dateline data source
from 2000 for observed travel behaviour. DATELINE suffers from some known problems e.g. with trip
frequencies and poor coverage in part of the EU. To overcome these problems we need to use
additional data sources in the modelling process.
In order to maintain consistency between dependent data (observed behaviour) and independent data
(Level of Service (LOS) and land use (LU)) that represents the situation in year 2000 is needed for the
estimation of long distance models. Background data will be delivered by an ongoing project ETIS+.
For long distance trips estimation, the following data are needed from ETIS+:

LOS variables. LOS data consists of two components: network data (the infrastructure) and
the traffic provided on the infrastructure i.e. travel times, frequency and capacity (?).

LU variables are variables describing the attraction of each zone as destination for trips.
Network data from 2001 is of poor quality which has been enhanced since then. In order to improve
model quality we will use LOS data from 2005 to approximate the situation in 2001. This is a departure
from our ambition of maintaining consistency despite the difference in time the 2005 data is regarded
as a better representation of the situation in year 2000 due to quality improvement. LOS may change
over time but usually quite slowly so we regard this as a minor problem compared to data quality in the
previous LOS data.
Additional data sources (Sigal, and others)
3.2 Data to support base case application
For the base case application year, data on population is needed by category in addition to LOS and
LU data. This will mainly be supplied by the ETIS+ project. Data on gross income would also be
desirable, as would more disaggregate data on car ownership. The car ownership issue is discussed
further in section 4.1.
3.3 Data to support estimation - short distance trips
According to the ETIS+ and Transtools3 DOW’s, base matrices by modes and travel purposes will be
established. These matrices can be used as aggregate demand. For this, population data by category,
LOS and LU data are needed for the base case year. This includes data for zone internal trips. This
data will also support the base case application.
3.4 Zoning system adjustment
The models are to be developed for the NUTS3 zone level. The current zones differ in size from just
over 10 000 inhabitants to more than 6 million inhabitants and this range is problematic to cover in a
model. In particular, congestion will be difficult to estimate correctly if the zones differ too much in size.
As discussed at the kickoff meeting, it is highly desirable to modify the zoning system to become more
consistent over countries. This will be handled within WP 5, with the aim of providing data delivered by
ETIS+ at the revised zone level. There is ongoing work with zone design and the new zone system will
have approximately 1600 zones.
3.5 Process ETIS+ (Rewrite wrt to what we will get from Otto concerning
variables etc)
It was agreed at the model design meeting that is desirable to receive the ETIS+ data successively as
it is ready, and not wait until all data is ready.
4. Modelling approach – some general issues
In this section we discuss the formulation of fundamental input variables such as car ownership, car
occupancy/cost sharing, vehicle fleet composition and work trip cost deduction. These variables are in
contrast to simple data (such as number of inhabitants) subject to a formulation process that depends
on the modeller and could as such be discussed.
4.1 Car ownership
Information on car ownership is essential for modelling the car mode. How the information is used will
depend on the type of information available. ETIS+ provides only an average number of cars for each
zone.
Passenger travel demand forecasting (including model estimation) requires assumptions or forecasts
on car ownership levels. Car ownership, or rather the vehicle stock, is included in the ETIS+ project
defined as follows:
“The data provided for the indicator ”vehicle stock” contains the numbers of different types of vehicles and is
differentiated by the application number of "passenger cars", "buses", "goods road vehicles", "motorcycles",
"special vehicles", "total utility vehicles", "road tractors", "trailers" and "semitrailers" in the respective region
(NUTS3).”
The current DoW does however not explicitly specify a car ownership model, and Transtools 3
forecasts would therefore have to rely on external data sources for car ownership projections. Another
issue is the vehicle fleet composition, which will have an impact on emissions as well as car running
costs. For Transtools 3 to be able to analyze policies in CO2 dimensions and to take account of car
ownership changes caused by policies or general economic growth, a consistent approach to car
ownership and vehicle fleet composition is needed.
The TREMOVE model is designed to consider car ownership effects and vehicle fleet composition.
When considering the use of TREMOVE models in Transtools 3, it appears that the treatment of car
ownership is quite a weak point. TREMOVE applies a “scrap and sales” cohort model for the vehicle
fleet composition, in which car ownership levels is an input that is needed to define the number of new
cars purchased in each year. This number is defined as (quote from the TREMOVE final report p 56):
“2. NEW SALES
From the demand module, we know the needed vehicle-km in year t. This is converted into the
number of vehicles needed to perform these, based on the average mileage of the vehicle category
(cars). This average mileage is calculated based on historic fleet and transport volume statistics23.
This number of vehicles is the desired stock. The difference between desired stock and surviving
stock are the sales of new vehicles (cars) in year t.”
The assumption of a fixed mileage per car is of course a strong assumption. Another problem is that
the forecasted total mileage (to be divided by the average mileages per car) is taken from the
SCENES model applied in the TREMOVE system. In the TREMOVE final report (p 222) the following
is said:
“The reference scenario in the TREMOVE demand module – called the baseline - is based on output
of the European transport model SCENES120 121. The run that has been used, is the P-scenario
from the ASSESS project, October 2005122.”
Looking at the ASSESS documentation (Final report, Annex VI p 19), it appears that car ownership is
an input to the SCENES model:
“The passenger demand model also requires a forecast car ownership per 1000 head in each of the
forecast years 2010 and 2020, for each EU25 country. These are based on national forecasts
collected by WSP during the TREMOVE 2 project in 2003 (TML, 2005; for car ownership data see
table below). Car stock forecast is built up based on data from Tremove at the country level. By using
the Eurostat Year 2000 car stock data and Tremove year 2020 car stock projections, year 2010 data is
estimated using linear interpolation. “
So, neither TREMOVE nor SCENES model car ownership. Consequently, Transtools 3 cannot rely on
these models to obtain car ownership forecasts. The treatment of car ownership as an exogenous
input is also a very weak part in the vehicle fleet composition model in the TREMOVE system. It is
therefore very desirable indeed to find a better solution to this problem, be it in the TREMOVE system
or in the Transtools system. In the final TREMOVE report different system improvements are
suggested (although not mentioning car ownership as one of these). In Annex F (Possible Approaches
To Revise The Demand Module In Tremove) to the Final report, the following option is outlined as an
alternative to revising the TREMOVE demand module:
“Given that both TRANS-TOOLS and TREMOVE operate at the EU scale and have been developed
on behalf of the Commission, it might be sensible to think of a possible integration of the two models.
Thus, instead of developing a new simplified demand generation module in TREMOVE it could be
chosen to use the capability of TRANS-TOOLS to model transport demand in detail and integrate in its
structure modules from TREMOVE.
Namely, three modules from TREMOVE could be integrated within the TRANS-TOOLS model:
- the vehicle stock module;
- the fuel consumption and emission module;
- the welfare module
In this perspective, the existing TRANS-TOOLS modules would be used to simulate transport demand
in detail (network level) while the specific features of the TREMOVE model – fleet development,
emissions, welfare computation - would continue to live within TRANS-TOOLS improving its
capabilities. Their development could continue in terms of “stand alone” modules (e.g. adding new
pollutants, improving vehicle choice algorithm, making scrapping rates dependent on energy price,
etc.) while, at the same time, gets the benefits of other developments of TRANS-TOOLS.”
Given the strong interdependence between passenger travel demand, car ownership and vehicle fleet
composition, it seems obvious that this should be handled within the Transtools framework. The
development of the necessary models and integrating the TREMOVE vehicle fleet composition model
in Transtools is however a major task that is currently not included in the Transtools budget. Beside
the crucial importance of car ownership in models of travel demand car ownership is subject to policy
decisions at different levels. The rate of growth in car ownership in Europe, particularly in regions
where the economy grows from low levels, has got a potential to be a very important change in the
European transport system.
The ASTRA model does contain a mechanism for car ownership forecasting at the NUTS 2 level. In
the ASTRA model, the ENV sub model generates the total car stock. This is done in the following way
(ASTRA Deliverable D4 p 114 ff):
“[i] Purchase Model for Development of Passenger Car Vehicle Fleet
Actually it is not the whole vehicle fleet that is calculated by the model but it is the changes of
the fleet between two time-steps that are caused by endogenous influences like personal
income, population density and exogenous influences like fuel price. In the past growing
disposable personal income was the major source for the increase of the car vehicle fleet.
ASTRA D4 ENVIRONMENT SUB-MODULE (ENV), Page: 115
While the increase in density has an counteractive effect. For regions with higher population
the vehicle fleet is smaller than for low densely populated regions with the same population.
This leads to the following basic equation:
VF = el_Inc * INC + el_PD * PD + el_FP * FP (eq. 9)
where: VF = change of vehicle fleet
el_Inc = fleet elasticity for income changes (>0)
INC = change of income
el_PD = fleet elasticity for changes of population density (<0)
PD = change of population density
el_FP = fleet elasticity for fuel price changes (<0)
FP = change of fuel price”
The elasticities used are presented as follows (ASTRA D4 Annnex A p 60):
12.4.9 Parameters for the Car Vehicle Fleet Model
The following table presents the elasticities that are applied in the vehicle fleet model
explained in chapter 6.4.3.1 of ASTRA D4. In the rows 2-3 the suggestions of
Johansson, Olof; Schipper, Lee (1997): “Measuring the Long-Run Fuel Demand of Cars”, in:
Journal of Transport Economics and Policy, Sep. 1997.are shown. However, currently the best fit for
the four macro regions is reached with the optimised values of row 5-8.
Table 73: Elasticities for the Car Vehicle Fleet Model
The REM submodel then distributes the stock on different subgroups (ASTRA D4 Annnex A p19 ff),
which are as follows:
“In each functional zone in the passenger model the population is segmented into 4 groups
based on
age and
economic position
The groups are:
Population under 16 (P1). - All persons under 16 years.
Population 16-64 Employed (P2). - This category includes all persons in full time and part
time employment.
Population 16-64 Not in Employment (P3). - This category includes all persons between
16 and 64 who are either unemployed or economically inactive.
Population over 64 (P4). - All persons over 64 years old.
At the same time the population of each functional zone is also segmented into three car
availability categories.
No car (C0) - Persons in households with no car
Part car (C1) - Persons in 2+ households with only one car i.e. part car
Full car (C2) - Persons in 1 adult households with 1+ cars and persons in 2+ adult
households which have 2+ cars, “
The ENV sub model also includes a distribution mechanism on vehicle type to reflect differences in
fuel consumption and emissions. Using the ASTRA capability would however be quite awkward when
running the Transtools system.
The level of ambition in treating the car ownership issue in Transtools has to be further discussed with
the commission.
4.2 Cost sharing car driver/car passenger
In order to correctly specify the travel cost by car we need to handle car occupancy. A simple way is to
divide the cost by the average car occupancy by trip type. Dividing the cost by car occupancy could be
overly simplistic since sharing car does not necessary mean shared costs. There is a dependency
between trip length and car occupancy. Longer trips are more likely to be done in groups which will
interact with a cost damping function. For long distance trips we will consider segmentation on party
size, which will be assumed to be exogenous to mode choice.
Short trips, long distance trips
4.3 Vehicle fleet composition
For getting LOS estimation data compatible with application data we need to apply the same
procedure as for application, i.e. a fuel cost based on fractions of car types with corresponding fuel
consumption rates and fuel prices. In previous Transtools versions, country specific fuel prices were
used because fuel taxation rules differ between countries.
Fuel costs were calculated per link and the cost for each link was based on the fuel cost of the country
the link falls within. It is important to separate the tax fraction from the raw fuel cost. If the taxation is
different from country to country, the relative effect of increased taxation will depend on the current
taxation offset.
4.4 Work trip cost deductions
Several countries apply tax deduction schemes for commute trips. This is the case in Belgium,
Denmark, Finland, France, Germany, the Netherlands, Norway, Sweden and Switzerland according to
a German study. In some cases there are special rules for commuting across borders. These
schemes are different and will have different impacts on trip distance and mode choice. This has to be
considered in order not to create bias in parameter estimation, and to allow for policy assessment. In
order to avoid bias in parameter estimation the costs used in the estimation phase must be calculated
taking tax deduction into account.
Depending on the formulation of the tax deduction scheme it may have several implications on other
policies that will be assessed using Transtools. The effect of monetary policies such as taxes, tolls and
fees could be less effective if tax deduction is allowed. Evaluation of pricing schemes will by erroneous
if tax deduction not is taken properly into consideration. The schemes are more or less complicated,
and it may be that some simplified way of handling this issue will be needed.
5. Short distance model
5.1 Scope
The scope is to establish a set of models by travel purpose containing mode, destination and
frequency choice. The model should be able to respond to transport policy changes with respect to
infrastructure and pricing as well as to background variables like land use, economic growth and car
ownership developments. It was noted during the meeting that car ownership is a difficult question in a
model that covers areas with large income differences. Since car ownership is a key variable in travel
demand models care is needed in formulation these variables. License holding is another variable that
often is present in models of travel demand together with car ownership for calculation of car
competition.
Commuting may take place over 100 km trip length. This may conflict with long-distance trips, if the
work purpose is defined in both categories. One option is to allow for longer work trips in the short
distance model and relax the upper limit of 100 km for that travel purpose. Estimating a separate
model for work trips in the long distance model will suffer from a low number of observed trips (about
300 in DATELINE).
Long commuting trips are typically less frequent than shorter ones. There are different options to
reflect different behaviour with respect to commuting distance. One is to introduce a choice between
long and short commute, the definition of which can be less than the 100 km criterion. Another option
is to use nonlinear utility function that will be more sensitive to the relatively few long distance
commutes. The definition of a commuting trip is not clear cut. Long distance commuting is often done
on a weekly basis where the commuter got a combination of an apartment close to work and house
more far away. The long commuting trip (Monday and Friday) will thus not be that burdensome and
show up in a non linear utility function in the cost dimension. In practise the long commute in the
example could or should be defined as leisure trip instead of commuting.
5.1.1 Assumed data availability:
A tour matrix (T) divided by mode (m) and purpose (p) from i to j on a day basis. In other words, the
matrix represents both the out-bound and the home-bound trip (i.e. a Generation –Attraction tour).
Level of service matrix (LOS) by mode (m), purpose (p) and type of time-component (k)
Monetary transport costs by mode (m) and purpose (p)
Car ownership and if possible licence holding by origin zone. .
Land use variable vector (LU) by destination,
Socioeconomic variable vector by origin (SE)
𝑝
𝑝
𝑝
i.e. 𝑇𝑖𝑗𝑚 , 𝐿𝑂𝑆𝑖𝑗𝑚𝑘 , 𝑇𝐶𝑖𝑗𝑚 , 𝐶𝑂𝑖 𝐿𝑈𝑗 , 𝑆𝐸𝑖
Some data could be difficult to obtain and estimates of LOS variables will probably be necessary. An
approach that could be used is to estimate transit speed e.g. with regard to presence of metro.
5.1.2 Modelling approach (Update wrt no slow modes and accommodating of
different tincome levels):
We start by taking a parsimonious approach, trying to avoid additional assumptions as much as
possible. Further discussion may modify the approach. We then specify a purpose specific model,
assuming a nested logit structure where destination is at the bottom, mode in the middle and
frequency at the top. This can of course be tested and changed if necessary, but will (hopefully) serve
as a base for initial model estimations. The utility functions at the different levels are specified below
(Greek letters are parameters to be estimated, C is a country or country group specific and P is a
purpose specific:
Destination:
𝐶𝑃
𝑃
𝐶𝑃
𝐶𝑃
𝐶𝑃
𝐺𝑇𝐶
𝑉𝑖𝑗𝑚
= 𝛽𝑚
∗ 𝑓(𝑇𝐶𝑖𝑗𝑚
, 𝐺𝐶𝑖𝑗𝑚
, 𝑤𝑘𝐶𝑃 ) + +𝛾𝑆𝐿
∗ 𝐵𝑜𝑟𝑑𝑒𝑟𝑖𝑗𝑆𝐿 + 𝛾𝑂𝐿
∗ 𝐵𝑜𝑟𝑑𝑒𝑟𝑖𝑗𝑂𝐿 + 𝛿 𝐶𝑃 ∗ 𝑆𝑖𝑧𝑒𝑗𝐶𝑃
_𝐶𝑃
𝐶𝑃
𝐶𝑃
𝐶𝑃
𝐶𝑃
Mode:
𝑉𝑖𝑚
= 𝐴𝑆𝐶𝑚
+𝜃𝑚
∗ 𝐿𝑜𝑔𝑠𝑢𝑚_𝑑𝑒𝑠𝑡𝑖𝑚 + 𝑎𝑚
∗ 𝐶𝑂𝑖
_𝐶𝑃
𝐶𝑃
𝐶𝑃
Frequency:
𝑉𝑖𝑡𝑜𝑢𝑟
= 𝑇𝑜𝑢𝑟_𝑐𝑜𝑛𝑠𝑡 𝐶𝑃 + ∑𝑆𝑠=1 𝜎𝑠𝐶𝑃 ∗ 𝑆𝐸𝑖𝑠𝑃 + +𝜃𝑚
∗ 𝐿𝑜𝑔𝑠𝑢𝑚_𝑚𝑜𝑑𝑒𝑖𝑚
The value-of-time is represented by 𝑤𝑘𝐶𝑃 and the representation of time and cost (in the 𝑓 –function)
may be specified in different flavours;
If represented in monetary terms the model would be
𝐾
𝑃
𝐶𝑃
𝑃
𝑓(𝑇𝐶𝑖𝑗𝑚
, 𝐺𝐶𝑖𝑗𝑚
, 𝑤𝑘𝐶𝑃 ) = 𝑇𝐶𝑖𝑗𝑚
+∑
𝑘=1
𝐶𝑃
𝑤𝑘𝐶𝑃 ∗ 𝐿𝑂𝑆𝑖𝑗𝑚𝑘
If formulated in time units it would be
𝐾
𝑃
𝐶𝑃
𝑃
𝑓(𝑇𝐶𝑖𝑗𝑚
, 𝐺𝐶𝑖𝑗𝑚
, 𝑤𝑘𝐶𝑃 ) = 𝑇𝐶𝑖𝑗𝑚
/𝑤𝑘𝐶𝑃 + ∑
𝑘=1
𝐶𝑃
𝑞𝑘𝐶𝑃 ∗ 𝐿𝑂𝑆𝑖𝑗𝑚𝑘
Where 𝑞𝑘𝐶𝑃 is an internal weighting of the different time components (take account of the fact that
congestion time may be weighted higher than free-flow time). From an estimation perspective the two
approaches are not different in a linear specification, however, implementation-wise they are as the
increase in 𝑤𝑘𝐶𝑃 will affect the forecast in different ways.
The two above forms are two extremes and the famous Train and McFadden paper discuss the
intermediate models and suggest that the time-unit model is the more likely (very small scale
estimation). Also the TT2 did apply the second version in the implementation but the first in the
estimation.
𝑆𝑖𝑧𝑒𝑗𝐶𝑃 is a composite destination size measure, defined as
𝑇
𝑆𝑖𝑧𝑒𝑗𝐶𝑃 = ∑
𝐶𝑃
exp(𝜏𝑡 ) ∗ 𝐿𝑈𝑡𝑗
𝑡=1
The data will facilitate estimation of an aggregate model. The dependent data will be market shares
corresponding to the ETIS+ data, associated with weights also given by the ETIS+ data. The shares
are found in the tour matrix elements fulfilling the conditions that the car distance is less than 100 km
or represent intra-zone traffic. As many zones will be relative large the only clear stratification if both of
these conditions apply. We realize that this may be rather crude for a number of matrix elements, but
ignore this as a first approximation.
The market shares for trip makers are defined as
𝑃
𝑇𝑖𝑗𝑚
𝑃
𝑃𝑖𝑗𝑚
=
𝑃𝑜𝑝𝑖
And for non trip makers
𝑃
𝑃𝑓=0
=1−
𝑃
∑ 𝑇𝑖𝑗𝑚
𝑃𝑜𝑝𝑖
where 𝑃𝑜𝑝𝑖 is the population in the relevant category.
The weights will be the population numbers in each origin (possibly for a given category).
This approach will enable estimation of the short distance model making the sole assumption of the
𝑤𝑘𝐶𝑃 factors corresponding to values of time for the various travel time components. These may be
obtained form Heatco and by making some additional assumptions for countries not included in the
Heatco work for the value of riding time, and by compiling information on aggress/egress time and
headway weights from other sources (such as the Wardman meta studies etc).
The choice between different sources of values of travel time components has been discussed during
project meetings. The values of HEATCO was questioned during our meetings and alternatives that
was suggested was to use values t derived by using proportionality to the wage rate or to use a PPP
index (Purchasing Power Parity-Index) . This proportionality to wage rate or PPP index can be derived
using values of time studies of known good quality (preferably more than one) and compute
conversion factors. ITS will look at this problem more in detail before a final decision is taken.
The model can be estimated simultaneously for the mode, destination and frequency choices. It can
also allow for different scales for different countries or country groups by separate estimation or scale
factors.
The Transtools 2 model used x modes (car as driver, car as passenger and public transport,). Slow
modes were not covered by Transtools 2, with regard to the zone size this is a reasonable solution.
Slow modes will most likely not be frequent choice in a model with the current scale for trips between
zones. For trips within zones slow modes is important but intra zone trips will not be assigned to the
network in the standard way. Still we think it is important to model intra zone trips in a reasonable
realistic way. For short distance trips intra zone trips will be the vast majority of trips and is important
to be able to calculate realistic trip rates regardless of zone size. Congestion will be depending on trips
with origin and destination within the same zone and in order to compute the number of cars that
contribute to congestion we thus think it is necessary to do some simplified mode split for intra zone
trips, including slow modes. It is also important to have a reasonable relation between the size of the
working population/number of work places and the number of generated work trips per zone,
perhaps in the generation step. Let the generation step be a choice between three choices: trip
outside the zone, trip within the zone, no trip. In the national Swedish transport model generation is
just the logsum from lower levels which is an alternative if zones are of reasonable similar size.
5.2 Calibration
This approach will also facilitate a simple calibration procedure, by adding as many calibration
parameters as wanted to the utility functions. The calibration will then simply be an additional
estimation on the same data, but by fixing the estimated parameters to the estimated values and
allowing for the estimation of calibration parameters.
Models will be estimated for four trip purposes:

Commuting



Private trips
Holiday trips
Business trips
6. Long distance models
The long distance models will have a similar structure as the short distance model but there will be
some differences that will be discussed in this section. In this section we take a look at data in the
Dateline survey and try to provide a background for a discussion on the following issues:




Segmentation with regard to journey duration and geography
Introduction of variables for barriers and affinities
Attraction variables
Use of seasonal dummy-variables
6.1 Data
The main data that will be used is the Dateline survey from 2001. There may be questions related to
this data source (Kay Axhausen). We will check (Jeppe) with Kay what these problems may be, and if
there exists a “cleaned” or pre-processed version of the data. We will also look at previous model work
(Hackney). To the extent it is possible to add other more recent data sources this will be done. Data
sources for the Danish National model as well as UK may be used. DTU will provide data related to
Denmark for further inspection.
In Table 1 we give an overview of the data in the Dateline survey. The total number of observations in
the data is sufficient for model estimation but as we will see below segmentation must be done with
some care. Of practical interest for the project is journeys and trips, excursions will not be modelled.
The number of commuting trips in the data is of limited use.
Table 1. Basic information about the Dateline survey.
Number of journeys
Number of trips
Number of excursions
Commuting
97 195 (60% domestic)
131 841
4 029
477
6.2 Segmentation by trip purpose
In the frequency table below we can see the number of journeys by purpose which sets a limit on
segmentation. About 55 % of all journeys in the material are holiday trips, another 35 % are other
private purposes and 10 % are business or work trips. The number of observations can be seen in the
table below.
Table 2. Journey by purpose in the Dateline survey.
Purpose
Holiday
Missing
Frequency
Percent
53316
54,9
Other
1068
1,1
Short holiday
1913
2,0
Visiting relatives/friends
5988
6,2
Leisure general
5053
5,2
Business/Work
9783
10,1
None
5497
5,7
Other
2757
,0
Total
85375
87,8
System
11820
12,2
97195
100,0
Total N of observations
About 55% of the journeys in the dateline survey were holidays. This calls for a rich description of
attraction variables related to holidays. Traditional “mass-variables” such as number of jobs or
inhabitants cannot explain the travel patterns of holidays where the purpose of the trip could be to go
to a place with nice climate.
Suggestions:
1. Winter holiday area, in the data numerous trips goes to small villages in the Alps. Without
special attraction it is impossible to understand why someone goes to small places like these.
2. Summer holiday area, see pt. 1 above
3. Temperature
4. Price level (in 2001)
5. Cultural heritage
One available variable that could be used as an attraction is the number of beds in hotels per zone. An
argument against that is that the number of beds is a response to demand rather than a pure
attraction variable. An argument to use beds as an attraction variable is that the number of beds is a
proxy for the attraction of the zone and can be regarded as exogenous from the travellers point of
view.
6.3 Seasonal variation of demand
Distribution of trips over the year is one issue that was discussed within the project group. Different
destinations will attract trips on a seasonal basis. Utilisation of the transport system will in areas with a
large tourism sector differ depending on the season. The same could be true for the level of service in
the public transport system where “seasonal peak hour traffic” could be the case. It will however be
computationally time consuming to do assignments for different seasons in addition to the
assignments for different time of day. Assignment in the long distance model will be done for an
average weekday (7 days).
16000
14000
12000
Other
10000
Holiday
Short holiday
8000
Vis rel, friends
6000
Leisure
4000
Business
2000
0
Jan Feb Mar Apr MayJune Jul Aug Sept Oct Nov Dec
Figure 1. Trip distribution over the year. Source: Dateline.
Seasonal variation is limited for all purposes except holiday trips.
6.4 Geographical segmentation
Geographical segmentation by region could be considered, e.g. southern Europe, northern Europe ...
Table 3 below shows the number of observations by country and gives some indication of what
segments that could be possible to use. The number of observations per country differs considerably
and indicates that segments by country must be made by groups if any.
Table 3. Number of observations by country of departure. Source: Dateline.
Type of journey
Business
Country code
Holliday
Private
Total
835
2095
3279
6209
of departure AS
0
1
0
1
AU
205
948
348
1501
BE
438
2134
872
3444
BO
0
1
0
1
BR
0
1
0
1
CE
0
0
1
1
DA
178
1417
508
2103
EI
75
326
158
559
EZ
0
0
1
1
290
1027
797
2114
FI
FR
2188
10036
4347
16571
GM
2380
9669
5094
17143
GR
358
2273
1493
4124
HU
0
1
1
2
ID
0
1
1
2
IS
0
1
0
1
IT
992
3328
1428
5748
LU
18
232
56
306
MN
0
2
1
3
NG
0
1
0
1
NL
717
3363
1371
5451
PL
0
3
0
3
PO
372
1852
1401
3625
SP
1144
10944
3449
15537
SW
511
1230
912
2653
SZ
91
749
189
1029
UK
1329
5039
2683
9051
US
1
6
1
8
VE
0
1
1
2
12122
56681
28392
97195
Total
Dateline contains no information about the organizational form of the journey e.g. charter, but contains
information on the number of non household members participating in the journey.
6.5 Non infrastructure network variables - Barriers and affinities
Different types of barriers and affinities could be considered in a modelling process, keeping in mind
that supply data probably are correlated with this kind of phenomenon. In order not to end up in an ad
hoc search for variables to enhance goodness of fit we rely on variables that have been used
successfully in previous studies. Examples of barriers and affinities that could be considered are:
national borders, language (common, similar or different), difference in costs, cultural
similarities/differences.
6.6 Trip duration
Segmentation on trip duration has turned out to be useful in the Swedish long distance model recently
estimated by Algers (2011). In the Swedish study the following segments (expressed in number of
nights away) were used: 0. 1-2, 3-5, 6+.
Main effects related to number of nights away from home:

Decreasing travel time sensitivity with regard to nights away




Decreasing importance of first wait time
Decreasing importance of travel cost
Increasing importance of summer house areas
Increasing importance for attraction variables associated with winter sports
Table 4. Trip duration (nights away) by type of journey in the Dateline survey.
Journey type
Business
Duration
Total
Holiday
Private
Total
,00
4712
2331
7498
14541
1,00
1545
0
2912
4457
2,00
1041
1
4834
5876
3,00
628
0
2985
3613
4,00
533
5712
532
6777
5,00
268
4078
252
4598
6,00
144
4074
111
4329
7,00
106
8691
187
8984
8,00
340
24446
492
25278
9317
49333
19803
78453
The number of observations for business trips limits the number of segments with regard to trip
duration. The number of observations by duration for holiday journeys raises some questions. The
difference between a holiday journey and a private journey, which isn’t too clear, could be a problem.
During our meeting it was suggested that holiday and private trips could by merged and then
segmented by duration.
6.7 Trip generation modelling
Commuting could be one trip purpose in the long distance model (see discussion above). An important
issue in long distance commuting is the frequency model. Frequency will depend on distance (time)
and we will need some model to estimate the development of long distance commuting over time.
6.7.1 Intra zone level of service
Intra zone traffic flows need to be modelled. One approach to obtain intra zone travel costs is use the
cost of travel to neighbouring zones*0.5 as an estimate. The estimate of intra zone traffic can then be
used to increase volumes on the network in order to obtain realistic levels of congestion..
6.8 Model for access and egress trips.
A long distance model will need support from a model of access and egress trips to the terminal of the
main mode of the trip. Most likely there will be a need to formulate different models for access and
egress trips respectively since the available modes will differ between the two trip types.
Access and egress trips will be included in the assignment routine and the work in WP8 will concern
formulation of a utility function of these trips. As for the long distance model we will depend on
Dateline but will also consider using national travel surveys with more detailed descriptions of long
distance trips and terminal trips.
7. Estimation procedures
7.1 Short distance trips
For short distance trips, estimation is reasonably straight forward. Software capable of estimating
aggregate nested models as well as simultaneously estimating composite size variables is sufficient
for this task. WP8 staff has considerable experience of using the Alogit software which meets these
requirements. The Alogit developer is also participating in the project, guaranteeing best possible
support.
Depending on how commuting will be defined with regard to the upper limit of trip length it might be
necessary to estimate nonlinear utility functions for short distance trips (cost damping). If commuting
only will be present in the short distance model it will be necessary to use a different cut off value for
these trips in the short distance model and consequently we must consider non linear utility functions.
In the next section on long distance trips different alternatives to estimate nonlinear utility functions will
discussed in detail. The reasoning for long distance trips also applies for short distance trips. If
nonlinear utility functions will be used or not is an empirical question, non linear functions could be
tested also for the short distance trip purposes that will be limited to < 100 km.
7.2 Long distance trips
For long distance trips, software capable of estimating nested models as well as simultaneously
estimating composite size variables for disaggregate data is also needed. An additional requirement is
the ability of estimating nonlinear utility functions. This can be done in different ways, having different
implications for resources needed and types of results obtained. The following approaches can be
defined.





Piecewise linear functions
Grid search procedures
Iterative search procedures
Direct estimation of form parameters (for example Box-Cox transformations)
Direct estimation of form parameter approximations
7.2.1 Piecewise linear functions
The possibility of defining nonlinearities by using piecewise linear functions is as old as the MNL
model. It has the advantage that it is simple, but the drawback of having to specify the intervals. The
approach also consumes one degree of freedom for each segment. A version of this approach is to
use information for the relative effects of the different segments to define a transformation of a
variable, and to estimate only one parameter for the piecewise transform. In the Swedish national
model Sampers this approach has been used for the headway variable, for which information on the
relative weights from a Stated Choice experiment for a set of headway intervals has been used to
define one single variable which is then piecewise nonlinear.
7.2.2 Grid search procedures
Another possibility is to define a continuous nonlinear transform, such as the Box-Cox transformation.
Then a set of models can be estimated for a corresponding set of transformation parameters, and the
optimal transformation parameter can be identified by comparing log likelihood values. An advantage
of this approach is that continuous nonlinear transforms are estimated, but the obvious drawback is
that a large number of runs is required, specifically if there are several nonlinear variables. The
procedure will give an estimate of the transformation parameters, but not of its standard deviation
which would also be useful to have. This approach has been partly used in the recent Swedish long
distance model research project.
7.2.3 Iterative search procedures
Instead of the grid search procedure in which the result of one model run is not used as input to
another run, a procedure that uses the outcome of a specific run to guess a better form parameter in
the next run can be used. In the recent Swedish long distance model research project, such a
procedure was implemented in the R environment using Alogit as a sub process.
7.2.4 Direct estimation of form parameters
Estimation of Box-Cox for parameters implies nonlinear utility functions and is therefore more difficult
to estimate. Dedicated software like Trio (Gaudry) and Biogeme (Bierliere) has been developed for
this task. While solving the problem of direct estimation of form parameters, other capability like
allowing for choices at different levels or composite size variables has been lacking. The added
complexity has also implied longer run times. This approach (using Biogeme 1.8) was tested in
context with the recent Swedish long distance model research project but was abandoned because of
run times.
7.2.5 Direct estimation of form parameter approximations
In a recent project for estimating long distance models for UK, approximations of form parameters
have been directly estimated. The approximation is defined by combining a logarithmic and a linear
term for the nonlinear variable and the resulting nonlinear variable is obtained by adding the two
together. The approach has been demonstrated to work well in the UK case. It was also successfully
tested in the recent Swedish long distance model research project, in which the approach has also
been generalised to include also negative form parameters (which cannot be estimated using log and
linear combinations). It must however be noted that the transformation parameter may have to be
constrained to the unit interval at least for some parameters. This generalisation and some Swedish
experience are included in the appendix.
7.2.6 Choice of approach
The approximation procedure may seem quite appealing. A possible drawback is that the form
parameter is not directly identified. Some tests (see appendix) suggest that the form parameter can be
retrieved from the approximation parameters, but probably without information on the standard
deviation. The retrieved parameter may however be used as a starting value in a direct form
estimation, thereby significantly reducing the run times.
There is also an ongoing software development process “out there”. A new version of Biogeme is
recently launched, allowing the user to compose his own likelihood function in Python, giving
additional flexibility and much faster performance. This version has not been tested in the recent
Swedish long distance model research project.
It is suggested that the approximation procedure is adopted in the Transtools project.
Appendix 1: Using Box-Cox approximations to estimate
nonlinear utility functions in discrete choice models
By Staffan Algers
Background
Discrete choice models are usually estimated using linear utility functions. Exceptions are models with
nested alternatives, and models using composite size variables which imply nonlinearity along certain
dimensions. But for central variables like cost and time components, a linear specification is usually
adopted. For quite some time it has however been argued that the linear specification should be
extended to allow for nonlinearities. Gaudry (2011) contains an overview and a compilation of
research efforts in this field. Recently attention has been given to this issue in the UK (the Cost
Dampening project) and in Sweden (High speed rail modeling project). Allowing for nonlinearities is
also a prerequisite for the models in the Transtools 3 project, which is the main reason for this paper.
Estimation of models with nonlinear utility functions obviously involves nonlinearities, which may make
the estimation more computationally burdensome. Nonlinearities, to the extent that they have been
introduced, have often been formulated as piecewise linear functions, or as predefined
transformations (like the logarithmic form or the square root). In this way functions involving nonlinear
parameters have been avoided – but at the price of not actually estimating the nonlinearity, only
testing specific nonlinear forms. A more general nonlinear formulation is the Box-Cox transformation,
which allows a continuous transformation as a function of a transformation parameter which can be
estimated simultaneously with the other parameters.
Dedicated software has been developed to tackle this estimation problem, like the TRIO program
(Gaudry 2008) and Biogeme (Bierlere 2003). These software tools are capable of estimating the BoxCox transformation parameter, but may have other limitations that make them less capable of handling
other estimation problems at the same time, like composite size variables or choices at different
choice dimensions (like using data for mode and destination choice where the destination is not known
for part of the data). The run time for these software tools may also be quite long, which is a
consequence of the more elaborate estimation procedure.
For estimation tasks involving large choice sets and complex nesting structures, there is a need for an
estimation process that is fast and feasible. Using approximations of the Box-Cox transformation has
been tested to some extent, and may be one interesting option. In recent modeling work for long
distance travel in the UK, combinations of linear and logarithmic terms were used to capture nonlinear
effects. This approach has been found to be useful also in the Swedish context. In this paper we will
describe how this option can be generalized and how it can be used in practical work.
The paper is organized in the following way: We first describe the Box-Cox transformation, and
possible approximations. Then we show an example of a real application of these approximations.
The Box-Cox transformation
Definition
The Box-Cox transformation is usually defined in the following way (Box and Cox 1964):
Special cases are when  is one or minus one, which corresponds to a linear function and the inverse
function. Another obvious special case is when  is zero, which corresponds to the logarithmic
function. In figure 1, transformations for some  values are shown.
Figure 1. Functional forms for different  values
Functional forms for different lambda
50
40
-0,5
30
-0,3
20
0
0,3
10
0,5
10
40
70
100
130
160
190
220
250
280
310
340
370
400
430
460
490
520
550
580
0
-10
In our applications, we would expect a negative parameter to be associated with the transformed
parameter, yielding a negative digressive function for parameters below 1.
Possible approximations
Assuming that the true functional form has the same general form as the Box-Cox transformation, we
want to find an approximation that will correspond to the unknown  value. Intuitively, one would think
that such a function could be obtained by a combination of two known functions – e.g. the function
with = 0.3 in figure 1 could be approximated by a combination of the functions with = 0.5 and  = 0
respectively. More generally this may be expressed as
𝑎∗
𝑦 𝜆 −1
𝜆
+ (1 − 𝑎) ∗
𝑦 𝜆+𝑑 −1
𝜆+𝑑
∼
𝑦 𝜆+𝑘 −1
𝜆+𝑘
where  is the combination parameter,  and d are known constants and k is the unknown parameter
that we want to obtain by identifying the combination parameter. The unknown k will be a function not
only of the combination parameter, but also of x – thus implying an approximation. We want the
approximation to be as close as possible, and for this we need a measure of the fit. The sum of the
squared differences is a standard measure of fit, which we adopt here as well. The combination
parameters can then be estimated by a linear regression on the two known transforms.
As the approximation depends on x, the combination parameter as well as the fit will then also be
depending on the range of x. In Figure 2, for a range of x values we have plotted a Box-Cox function
with a 0.5 transformation parameter and a utility parameter equal to minus one. We have also plotted
the corresponding functions obtained by linear regression of the logarithm ( = 0) and the linear ( = 1)
transformations in two partly overlapping ranges. The Range 1 function has been estimated for x
values between 20 and 620, whereas the Range 2 function has been estimated for x values between
120 and 720. The plots are all in the 20 – 720 range.
Figure 2
0
20 60 100 140 180 220 260 300 340 380 420 460 500 540 580 620 660 700
-10
-20
BC(X)
-30
Approx_Range 1
Approx_Range 2
-40
-50
-60
This is an illustration of the nature of the approximation. As can be seen, the approximation is quite
good in the range for which it is estimated, but deviates notably outside this range. This can be further
illustrated by looking at the residuals in figure 3.
Figure 3
2
1
0
20 60 100 140 180 220 260 300 340 380 420 460 500 540 580 620 660 700
-1
Diff_Range 1
-2
Diff_Range 2
-3
-4
-5
The R2 for Range 2 is larger than for Range 1 (0,99974 and 0,99903 respectively). The fit to data
within each range is obviously quite good, but will be worse if the estimated functions are extrapolated,
specifically in the lower range (where the slope of the function changes faster).
Using the logarithmic and the linear functions as known functions has been applied in real
applications. Daly (2011) also has shown the equivalence between the Box-Cox function and the
gamma function, see appendix 2. But this may not be the optimal approach. Intuitively, we would
expect a closer approximation if we use known functions that are closer to the true function. In figure 4
we again plot the true Box-Cox function having a transformation parameter of 0.5 and a utility
parameter of minus 1, together with the two estimated approximations, now using Box-Cox functions
with transformation parameters 0.7 and 0.3.
Figure 4
0
20 80 140 200 260 320 380 440 500 560 620 680
-10
-20
BC(X)
-30
Approx_Range 1
Approx_Range 2
-40
-50
-60
It is now very difficult to observe any differences. The fit of the two approximations has now increased
to an R2 of 0.999969 and 0.999992 for Range 1 and Range 2 respectively. The residuals are plotted
in Figure 5.
Figure 5
0.4
0.2
20
60
100
140
180
220
260
300
340
380
420
460
500
540
580
620
660
700
0
-0.2
Diff_Range 1
Diff_Range 2
-0.4
-0.6
-0.8
The pattern of the differences is still the same, but reduced in size. So, by using functions that are
closer to the true function, we may achieve closer approximations. It also turns out that using two
closer functions is more important than using functions which are embedding the true function. In
figure 6 we plot approximations using known functions with transformation parameters 0.7 and 0.6.
Figure 6
0
20 80 140 200 260 320 380 440 500 560 620 680
-10
-20
BC(X)
-30
Approx_Range 1
Approx_Range 2
-40
-50
-60
These functions perform even better than using the 0.7 and 0.3 transformation parameters. The
residuals are now even smaller (Figure 7), which depends on the fact that we now use known
functions that are now even closer to the true function (the transformation parameters 0.7 and 0.6 are
closer the true transformation being 0.5).
Figure 7
0.35
0.3
0.25
0.2
0.15
Diff_Range 1
0.1
Diff_Range 2
0.05
0
-0.05
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
-0.1
-0.15
Another limitation of the approach with the logarithmic and the linear approach, i.e. using the
transformation parameters 1 and 0, is that it is difficult to approximate true functions with negative
transformation parameters. In figure 8 we plot a true Box-Cox function with the transformation
parameter -0.5, and approximations using the 0 and 1 (Approx_Range 1) respectively 0 and -1
transformation parameters (Approx2_Range 1). For simplicity, we use only the lower data range
(Range 1).
Figure 8
-1
20 60 100 140 180 220 260 300 340 380 420 460 500 540 580 620 660 700
-1.2
-1.4
BC(X)
-1.6
Approx_Range 1
Approx2_Range 1
-1.8
-2
-2.2
If the known functions with transformation parameters 0 and 1 are used, the regression parameters
get different signs. This implies that the resulting approximation is no longer monotonically decreasing,
as the positive component eventually will more than offset the negative component. This is of course a
very undesirable property, which has prohibited the use of this particular approximation in the negative
range of the Box-Cox transformation parameter. The approximation is still fairly close in terms of R2
(0.99757), but the approximation using 0 and -1 as transformation parameters for the known functions
is closer, and does not have the undesirable property.
As before, we may inspect the residuals for the different approximations (figure 9):
Figure 9
0.5
0.4
0.3
Diff_Range 1
0.2
Diff_Range 2
Diff2_Range 1
0.1
Diff2_Range 2
0
20 60 100 140 180 220 260 300 340 380 420 460 500 540 580 620 660 700
-0.1
-0.2
In Figure 9, residuals from four approximations are plotted. The 0/1 transformation approximations are
denoted Diff, and the 0/-1 transformation approximations are denoted Diff2. It is quite obvious that the
0/-1 parameter approximation performs better than the 0/1 approximation parameters in both ranges,
giving an R2 of 0.99994 and 0.99999 for the low and high ranges compared to 0.99757 and 0.99968
for the 0/1 transformation parameter approximation.
As for positive true transformation parameter cases, we can improve the approximation by using
closer known functions. In figure 10, we plot the residuals for the 0/-1 transformation parameter
approximation (labeled Diff) as well as for a -0.3/-0.7 transformation parameter approximation (labeled
Diff2).
Figure 10
0.04
0.02
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35
-0.02
Diff_Range 1
-0.04
Diff_Range 2
-0.06
Diff2_Range 1
Diff2_Range 2
-0.08
-0.1
-0.12
-0.14
It is obvious from the figure that the -0.3/-0.7 transformation parameter approximation gives a closer
approximation, especially when extrapolating the approximation.
It is of course desirable to obtain as close an approximation as possible. As has been shown, the
approximation can be improved by closing in on the true (but unknown) transformation. A question is
then how we can know how to change the known function transformation parameters to become
closer to the unknown transformation parameter. One way is of course to plot the known functions and
the estimate using a wider transformation parameter range for the known functions (such as the 0/1
range). Another way would be to reverse engineer the regression parameters to obtain the Box-Cox
parameters that minimize the (squared sum of) the residuals between a Box-Cox function and the
estimated function value. That will give a transformation parameter value that can be used to guide the
choice of transformation parameters for the known functions.
Using approximations in real applications
Up to this point we have examined data without any disturbances, and not really in a discrete choice
setting. In real applications however, we will use approximations in a discrete context where random
disturbances and correlation among different variables will influence the estimates. We first note that
real applications of the method has been used in modelling work in the UK (Fox et al 2009), who used
the 0/1 transformation parameter combination for the known functions. In recent Swedish research,
also the 0/-1 transformation parameter combination was used. In this research, mode and destination
models were estimated for tours longer than 100 km single trip. The tours were segmented according
to purpose and duration, and in this section we will use the model for mode and destination for private
tours arriving and departing from the destination on the same day.
The model structure was a nested logit model, having mode choice at the top and then a destination
level at the municipality level (290 municipalities in Sweden) in the middle and a second destination
level at the bottom with 670 zones. To keep the number of alternatives to a reasonable size, 20
municipalities were sampled at random, and then 2 zones were sampled from each municipality. The
modes comprised car, bus, train and air.
For this data set, the 0/1 approximation was first applied. The results are shown in table 2 together
with a linear model version. For simplicity, a number of variables (mostly dummy variables associated
with different modes) are excluded from the table.
Table 2
Name
Observations
Final log(L)
D.O.F.
Rho²(0)
AEA
AccEgrBT
LinFW
LinTC
LogTTBA
LinTTBA
LogC_0
LinC_0
LogC_12
LinC_12
LogC_34
LinC_34
LogC_56
LinC_56
SizeCS
SizeSH
Theta1
Theta2
TT_C_A1
TT_C_A2
FW_A1
FW_A2
Linear
4320
-9586,0713
33
0,5390
-0,02613
-0,03598
-0,00207
-0,02782
0,00000
-0,01170
0,00000
-0,01077
0,00000
-0,00737
0,00000
-0,00550
0,00000
-0,00370
1,00000
0,19361
0,69854
0,78149
t-value
(-4,4)
(-9,6)
(-2,4)
(-24,8)
(*)
(-14,3)
(*)
(-9,8)
(*)
(-9,4)
(*)
(-7,9)
(*)
(-2,6)
(*)
(2,3)
(38,0)
(14,5)
First approximation
4320
-9035,4795
40
0,5655
-0,02475
-0,04267
t-värde
t-value
(-3,8)
(-10,3)
Second approximation
4320
-8994,9004
40
0,5675
-0,02501
-0,04201
0,00000
-1,00381
-0,00296
-2,10315
-0,00308
-2,31340
-0,00092
-2,17654
-0,00065
-1,33745
-0,00126
1,00000
0,26786
0,72459
0,56673
-3,96185
0,00747
-0,68848
0,00335
(*)
(-3,5)
(-2,6)
(-6,4)
(-2,3)
(-8,0)
(-1,2)
(-7,7)
(-1,0)
(-2,6)
(-0,7)
(*)
(3,1)
(39,2)
(11,4)
(-13,1)
(6,8)
(-4,4)
(2,2)
0,00000
-1,10896
-0,00224
-1,95771
-0,00394
-2,19586
-0,00144
-2,08627
-0,00106
-1,25979
-0,00178
1,00000
0,30440
0,74810
0,53996
-0,05646
-343,49939
-0,47483
2,26343
(*)
(-4,0)
(-2,1)
(-6,2)
(-2,8)
(-7,9)
(-1,9)
(-7,7)
(-1,6)
(-2,5)
(-1,0)
(*)
(3,5)
(40,1)
(11,7)
(-0,2)
(-10,7)
(-3,2)
(0,5)
(-3,9)
(-10,2)
The parameters reported here concern access/egress time and headway for public transport, separate
in vehicle times for car and public transport, income segment specific cost, zone size and the nesting
parameters. All these are significant at normal risk levels in the linear model. Allowing for
nonlinearities by adding a logarithmic term for headway, in vehicle times and costs improves the
model quite a lot in terms of log likelihood units. For the public transport in vehicle time variables, the
two parameters appear negative which suggests transformation parameters in the unit interval. For the
cost variables this is also the case, but some of the linear components are not statistically significant.
For headway and car time, the linear components are positive, suggesting that the true functions have
negative transformation parameters. As an illustration, we plot the estimated approximation of the
headway (Y) as well as the closest possible Box-Cox transformation Est(BC(X)) over the value range
in the data(Figure 11). It must however be noted that the transformation parameter may have to be
constrained to the unit interval at least for some parameters (see Daly 2010).
Figure 11
10
40
70
100
130
160
190
220
250
280
310
340
370
400
430
460
0
-0.5
-1
-1.5
-2
Y=F1par*F1 +
F2par*F2
Est(BC(X))
-2.5
-3
-3.5
The problem with the approximation is obvious. The Box-Cox transformation parameter giving the best
fit has the value -0.37. This suggests that the approximating functions should be closer to this value.
Correspondingly, the in vehicle car time transformation giving the best fit to a Box-Cox function has
the value -0.08. We therefore change the transformation parameters of the known functions from 0
and 1 to 0 and -1 for both headway and car in vehicle time. We then get the results labelled “Second
approximation” in Table 2.
We now get another improvement in terms of log likelihood.
References
Bierlaire, M. (2008). "An introduction to Biogeme", http://biogeme.epfl.ch/
Box, G.,E.;P. and Cox, D.R. (1964) An Analysis of Transformations. Journal of the Royal Statistical
Society. Series B (Methodological). Vol 26, No2 pp 211 – 252
Daly, A. (2011) Equivalence of Box-Cox and ‘Gamma’ functions, unpublished note
Fox, J., Daly A., Patruni B. (2009) Improving The Treatment Of Cost In Large Scale Models Paper
presented at the 2009 ETC conference
Gaudry, M. (2008) Non linear logit modelling developments and high speed rail
profitability, Agora Jules Dupuit, Publication AJD-127, www.e-ajd.org.
Gaudry, M., Duclos, L-P., Dufort, F., Liem, T. (2008) TRIO Reference Manual, Version 2.0
http://www.e-ajd.net/
In the following examples of tables, figures and formulas are provided with headings and heading
numbers which must be updated in order to provide the right number.
Appendix 2 Equivalence of Box-Cox and ‘Gamma’
functions
Andrew Daly, RAND Europe, 2 November 2011
This note explores the equivalence between Box-Cox and ‘Gamma’ functions and their use in choice
modelling.
Basic equations
Box-Cox is defined as usual by
( )
x  1
,
when   0
x ( )  log  x ,
when   0
x


Inspired by old-fashioned transport planning jargon, we use the ‘Gamma’ label to apply to functions
defined by1
𝑥 [∝] =∝ 𝑥 + (1−∝)𝑙𝑜𝑔(𝑥)−∝
Note that when x=1, both transformations take the value 0, whatever the value of . Moreover, if we
differentiate
′
𝑥 (∝) = 𝑥 ∝−1
′
𝑥 [∝] =∝ +(1−∝)𝑥 −1
both of which are equal to 1 at x=1, whatever the value of . Finally, when we differentiate a second
time, we obtain
′′
𝑥 (∝) = (∝ −1)𝑥 ∝−2
𝑥 [∝] ′′ = −(1−∝)𝑥 −2
both of which are equal to (-1) at x=1.
So at this specific value of x, the functions are equal and have the same slope and curvature. It also
seems that the function of  is the same in each case.
1
More standard mathematical nomenclature is to define the gamma function as the integral of the exponential of this function
(see http://en.wikipedia.org/wiki/Gamma_function). The subtraction of  in this formula makes the functions equal at x=1.
However, the question is what happens when we move away from x=1. Clearly this depends on the
value of : this is explored in the following graphs, which show first the transformed values for
=0.333... and =0.666... for values of x from close to 0 to 5; then the transformed values for x=0.2,
x=0.5, x=2 and x=5 for the full range of  values between 0 and 12.
Looking first at the curves for fixed , we see that for high and low values there is some difference, but
for a substantial range around 1 the curves are very close.
Difference between Gamma and Box-Cox at
different values of 
transformed x value
3
2
1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
-1
-2
-3
-4
x
BC (a=0.33)
Gamma (a=0.33)
BC (a=0.67)
Gamma (a=0.67)
Looking at the curves for fixed x, we note that for =0 and =1 the functions are identical, so it is only
for intermediate values that there is a difference. For x=2 the maximum difference occurs at x=0.52
and is about 2%, too small to be clear on the graph. For x=5, the maximum difference is also at
x=0.52 and is about 13%. For x=0.5 we have a maximum difference less than 2%, but for x=0.2 the
difference is as much as 8%. Again, for x close to 1 the differences are small, but for large and small
values larger differences occur but with the range of a factor of 5 larger or smaller the differences are
not large.
Values of  outside this range are not compatible with the framework of the problem; see Daly, A.
(2008) The relationship of cost sensitivity and trip length,
2
http://www.dft.gov.uk/pgr/economics/rdg/costdamping/pdf/costdamping.pdf.
Difference between Gamma and Box-Cox at
different values of x
transformed x value
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1
-2

BC (x=2)
Gamma (x=2)
BC (x=5)
Gamma (x=5)
BC (x=0.2)
Gamma (x=0.2)
BC (x=0.5)
Gamma (x=0.5)
Conclusion
We conclude that for values of x close to 1 these transformations are performing the same function
and are closely equal. Further from 1 the differences are not so small, but they are still not large.
Moreover, simply by using the same parameter values (noting the subtraction of  from the Gamma
formulation) we can get exact equality when x=1 and when =0 or =1. The parameter  has
essentially the same meaning in the two functions.
Use in choice modelling
In choice modelling using RUM the Gamma formulation is much easier to use than Box-Cox as we
can set it up as a linear-in-parameters model of utility
𝑉 = ⋯ + 𝛽1 𝑥 + 𝛽2 𝑙𝑜𝑔(𝑥)
with ∝= 𝛽1 ⁄(𝛽1 + 𝛽2 ). To maintain the range of  in [0, 1] we need not to have  parameters with
opposite signs, which is necessary in any case to fit with the logic of utility functions.
ALOGIT is not able to estimate directly models that are not linear in parameters; while Biogeme can
do this, it tends to be very slow in converging, so in this case also the Gamma formulation is
preferable.
We showed close equivalence between Gamma and Box-Cox for values of x close to 1. In practice
this can be achieved by using a value in the modelling divided by the mean, so that a clustering of
values close to 1 would be expected and values such as 5 might be unusual.
It would be interesting to make some estimations of Gamma models and then to repeat the runs using
Box-Cox with the same formulation to see how similar the results appeared to be.
In conclusion, the Box-Cox and ‘Gamma’ transformations give very similar results when the same
transformation parameter is used. There is no reason to regard one of these functions as having
higher status than the other, so that we may use whichever is more convenient. For choice model
estimation this is the gamma function.